The $L^2$-boundedness conjecture for Grushin pseudo-multipliers

Let $G$ be the Grushin operator, let $\boldsymbol{L}$ and $\boldsymbol{U}$ denote the associated operator-valued spectral variables, and let $\mathcal{S}^0_{\rho,\rho}(\boldsymbol{L},\boldsymbol{U})$ be the corresponding symbol class. For $0\leq\rho<1$, consider the pseudo-multiplier $m(x,\boldsymbol{L},\boldsymbol{U})$ associated with a symbol $m\in\mathcal{S}^0_{\rho,\rho}(\boldsymbol{L},\boldsymbol{U})$. The $L^2$-boundedness conjecture. Let $m \in \mathcal{S}^0_{\rho, \rho} (\boldsymbol{L}, \boldsymbol{U})$ for some $0 \leq \rho < 1$. Then the operator $m(x, \boldsymbol{L}, \boldsymbol{U})$ extends to a bounded operator from $L^2(\mathbb{R}^{n_1 + n_2})$ to itself. The authors explain that their proof of the relevant theorem requires a cancellation condition, but they believe the stated $L^2$-boundedness should hold without that assumption; its resolution is not established in the supplied text.